I agree Kirby, upfront definitions are appreciated by all - teachers, students AND curriculum writers. But there's an awful lot of confusion possible if definitions are purely localized. Speaking from an Australian elementary/middle-school perspective, which is not too dissimilar to the USA, there are multiple definitions that students may have the contend with.
We now have a national curriculum which does not state the definitions for quadrilaterals. If this document replaces the states' individual curricula then somewhere along the line definitions will need to be provided by someone - most likely the state education bodies. So then the state education bodies may differ in definitions which doesn't help students who move states, nor is especially useful in developing consensus on how to teach this aspect of the curriculum.
If the state bodies don't define them, the textbook writers will. Given that upper grades in the same school may use different textbooks to those in the lower grades, students get confused even staying within the same school. Assuming everyone uses the same textbook in a school, the school down the road may not and the mobile students get lumped with the problem again. If the textbook writers don't define them, the teacher will and will have to find (hopefully) a range of definitions, some of which are incomplete, inconsistent or conflicting, in dictionaries or similar then decide which ones to use and why.
On top of all of this there is national testing. Every teacher and every school can localize themselves until they're blue in the face but if the test writers are using their own special set of definitions many kids are going to fail that part of the test.
In my non-working life I don't give two hoots about quadrilaterals and get by with the names I learnt in pre-school. Given the blank stares I get from anyone who is not a teacher (and many who are) when I talk about different quadrilaterals I safely assume that I'm not alone. If learning the names and properties of quadrilaterals has any value at all in a curriculum I think they need more thought than has been given them.